Composite bound states and broken U(1) symmetry in the chemical-master-equation derivation of the Gray-Scott model.

We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): U+2V →[λ]3V and V → [μ]P, U → [ν]Q, with a constant feed rate for U. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model, which now involves two composite fields and also intrinsic noise terms. One of the composites is ψ(1) = φ(v)(2), where {φ(v)}(η) =v is the concentration of the species V and the averaging is over the internal noise η(u,v,ψ(1)). The second composite field is the product of three fields χ = λφ(u)φ(v)(2) and requires a noise source to ensure probability conservation. A third composite ψ(2) = φ(u)φ(v) can also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero-energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields φ(u), φ(v),χ that induce higher-order processes such as n → n scattering for n>3. The amplitude of the noise acting on φ(v) is itself stochastic in nature.